Generalized coordinates

Generalized coordinates

In this lesson, we'll give a brief explanation of what generalized coordinates are. Up till this point, we got away with a very vague and imprecise definition. But for some of the problems that we'll look at later (such as the double-pendulum problem), the more technical definition we develop in this lesson will be very useful.

Derivation of Guass's Law from Column's Law

Derivation of Guass's Law from Column's Law

Guass's law: for any charge distribution, whether a system of a finite number of point charges or a continuous distribution of charges, if I draw any arbitrary closed surface (which can be of any shape, size, and location), then the only contributions to the electric flux through that closed surface will be due to only the charges \(q_{enc}\) enclosed within that surface and that electric flux is given by the amount \(\frac{q_{enc}}{\epsilon_0}\). It is important to appreciate the word any to fully grasp the applicability of Guass's law: it applies to any closed surface and to any distribution of charge.

Column's Law

Column's Law

Column's law describes the electric force between any distribution and number of charged particles. It is derived empirically and falls off inversely with the square of the distance, analogous to Newton's law of gravity. The practical usefulness of this law is that it allows one to determine the effect that any distribution of charge has on any other distribution of charge; this is analogous to how Newton's law of gravity allows one to determine the effects of one mass distribution on another. One very gradually learns this by solving many problems. But in this section, we'll start off with Column's law and then proceed to do some algebra and calculus in order to derive equations which describe the electric force that systems and continuous charge distributions exert on an individual particle.

Derivation of the Euler-Lagrange Equation

Derivation of the Euler-Lagrange Equation

In this section, we'll derive the Euler-Lagrange equation. The Euler-Lagrange equation is a differential equation whose solution minimizes some quantity which is a functional. There are many applications of this equation (such as the two in the subsequent sections) but perhaps the most fruitful one was generalizing Newton's second law. This equation, together with Hamilton's principle, allows us to generalize Newtonian mechanics and find the motion of systems using generalized coordinates—all of a sudden, double-pendulums don't seem so scary.

Electric Flux

Electric Flux

Unlike the flux of the velocity field of a fluid, the notion of electric flux is much more abstract. Like concepts such as energy, it is very abstract but nonetheless very useful in analyzing certain situations. In this section, we derive the electric flux of any electric field through any arbitrary surface. Only later one, when we discuss Guass's Law, will we learn about the usefulness of this idea.

Noether’s Theorem

Noether’s Theorem

Noether's Theorem is one of the most profound theorems in all of science, let alone physics. It implies that the very fabric of space and time give rise to the laws of nature and lead the mathematician David Hilbert to independently discover the general theory of relativity, alongside Einstein. Noether's theorem, in simplest terms, is the statement that for every symmetry (what a symmetry is is something we'll get to in this lecture) there is an associated conservation law. Our goal in this lecture will be to prove that and show that various different conservation laws follow from an associated symmetry.